{"paper":{"title":"Directional Curvature from Armijo Backtracking: A Low-Cost Sharpness Probe and a Calibration-Free Learning-Rate Safeguard for Adam","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.LG","authors_text":"Ashmitha R, J\\\"org Frochte","submitted_at":"2026-07-04T19:49:01Z","abstract_excerpt":"The local sharpness of the loss, the top Hessian eigenvalue $\\lambda_1$, determines the largest stable gradient step, but measuring it normally requires Lanczos or Hessian-vector iterations. We observe that a single Armijo backtracking line search already carries this information at the cost of a few forward passes: the accepted step $\\alpha$ brackets the \\emph{directional} curvature $q = g^\\top H g/\\|g\\|^2$ within the multiplicative band set by the backtracking factor. Across CIFAR-10, Fashion-MNIST and Imagenette, $\\log\\alpha$ tracks $\\log\\lambda_1$ at Pearson $-0.91$ to $-0.95$, giving a lo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.03998","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.03998/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}